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Published February 1, 2019 | Supplemental Material
Journal Article Open

Enhanced local maximum-entropy approximation for stable meshfree simulations


We introduce an improved meshfree approximation scheme which is based on the local maximum-entropy strategy as a compromise between shape function locality and entropy in an information-theoretical sense. The improved version is specifically designed for severe, finite deformation and offers significantly enhanced stability as opposed to the original formulation. This is achieved by (i) formulating the quasistatic mechanical boundary value problem in a suitable updated-Lagrangian setting, (ii) introducing anisotropy in the shape function support to accommodate directional variations in nodal spacing with increasing deformation and eliminate tensile instability, (iii) spatially bounding and evolving shape function support to restrict the domain of influence and increase efficiency, (iv) truncating shape functions at interfaces in order to stably represent multi-component systems like composites or polycrystals. The new scheme is applied to benchmark problems of severe elastic and elastoplastic deformation that demonstrate its performance both in terms of accuracy (as compared to exact solutions and, where applicable, finite element simulations) and efficiency. Importantly, the presented formulation overcomes the classical tensile instability found in most meshfree interpolation schemes, as shown for stable simulations of, e.g., the inhomogeneous extension of a hyperelastic block up to 100% or the torsion of a hyperelastic cube by 200° –both in an updated Lagrangian setting and without the need for remeshing.

Additional Information

© 2018 Elsevier. Received 25 April 2018, Revised 18 October 2018, Accepted 21 October 2018, Available online 29 October 2018. DMK and SK acknowledge support from the National Science Foundation through CAREER Award CMMI-1254424. KD acknowledges support from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant agreement no. 636903).

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